Chapter 1: Functions and Their Graphs

Today in class, we reviewed shifting, reflecting, and stretching/shrinking.

At this point in the trimester, our class has only began to scratch the surface of the importance of functions and their graphs. We have already discussed the basics of functions: how functions are defined, what they look like, the domains and ranges of these function, and how to graph simple functions. Today, we reviewed a familiar, but extremely important aspect of functions-- how to manipulate them.


For each manipulation, consider this parent function and its graph:

SHIFTING:

In order to vertically shift any graph (either up or down), remember the equation:



**C, a constant, affects the output value in this equation, not the input value**

C > 1  Graph shifts upwards

C < 1  Graph shifts downward

EXAMPLE: Shiftup two units. Graph the parent function in blue and the new function in green

In order to horizontally shift any graph (either left or right), remember the equation:



**C affects the input value in this equation, not the output value**

C is a positive number  Graph shifts to the left

C is a negative number  Graph shifts to the right


EXAMPLE: Shiftleft two units. Graph the parent function in blue and the new function in green

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  

REFLECTING:

In order to vertically reflect any graph across the x-axis, remember the equation:




*Because the negative is applied to f(x), only the output values are affected whereas the input values remain the same*

EXAMPLE: Reflectacross the x-axis. Graph the parent function in blue and the new function in green

*Notice how the inputs (x values) remain the same while the outputs (y values) are multiples by -1*


In order to horizontally reflect any graph across the y-axis, remember the equation:



*Because the negative is applied to (x), only the input values are affected whereas the output values remain the same*

EXAMPLE: Reflectacross the y-axis. Graph the parent function in blue and the new function in green

WHY DOES THIS GRAPH LOOK THE SAME AS THE FUNCTION?! Because reflecting the parent function over the y-axis yields the exact same graph :)



- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  

STRETCHING/SHRINKING:

Remember, stretching and shrinking distorts the graph (changes its original shape)!

In order to vertically stretch/shrink any graph, remember the equation:





**C affects the output value in this equation, not the input value**

C > 1  Vertical stretch-- getting closer to the y-axis

0 < C < 1  Vertical shrink-- getting farther away from the y-axis

Vertical Stretch: by a factor of 2

Vertical Shrink: by a factor of 1/2

In order to horizontally stretch/shrink any graph, remember the equation:


**C affects the input value in this equation, not the output value**

C > 1  Horizontal shrink-- getting farther away from the x-axis

0 < C < 1  Horizontal stretch-- getting closer to the x-axis


Horizontal Shrink: by a factor of 2

Horizontal Stretch: by a factor of 1/2

OVERALL: 

Perhaps the most important equation you can take away from this lesson is this one:

a= Vertical stretch/shrink (stretch: c>1 shrink: 0<c<1)

b= Horizontal stretch/shrink (stretch: 0<c<1  shrink c>1)

c= Horizontal translation (left or right)

d= Vertical translation (up or down)

This equation sums up everything we discussed in today's class in a neat and concise manner!

Hope this helps! Good luck in Honors Pre-Calc :)